Journal
ANNALS OF APPLIED PROBABILITY
Volume 29, Issue 5, Pages 3155-3200Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/19-AAP1477
Keywords
Stochastic Volterra equations; Riccati-Volterra equations; affine processes; rough volatility
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Funding
- AXA Investment Managers
- Swiss National Science Foundation (SNF) [205121_163425]
- Chair Markets in Transition (Federation Bancaire Francaise)
- Universite Paris Dauphine
- [ANR 11-LABX-0019]
- Swiss National Science Foundation (SNF) [205121_163425] Funding Source: Swiss National Science Foundation (SNF)
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We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.
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